4 THE VAGUE TOPOLOGY 109 (ii) The vague closure of H in MC(X) is a compact metrizable space with respect to the vague topology. (i) This is an immediate consequence of the Banach-Steinhaus theorem (12.16.4). (ii) Let (Uw) be a sequence of relatively compact open sets in X which cover X and are such that €"„ c UB + 1 (3.18.3). Since every compact subset K of X is contained in some Urt (3.16), each space Jf (X; K) can be identified with a closed subspace of one of the Banach spaces #(On), hence with a closed subspace of Jf (X; On). Now we know (7.4.4) that «*((}„) is separable, hence the same is true of jf (X; On) (3.10.9). Let (fmn)m^ be a dense sequence in ^(X; Uw). To show that H is metrizable it is enough to show that the vague topology on H can be defined by the pseudo-distances |</mn, /i — v>| (12.4.6). This means that if gi (1 ^ i ^ p) is a finite sequence of functions belonging to ;fc(X), if jU0 is an element of H and r a real number >0, there exists a Unite number of functions fmknk (l^k^q) such that the relations /*6 H ud \<fmknk,H-»o>\^$r(l ^k^q) imply |<^, ji - ^0>| g r (1 £/£/>). But the #f all belong to Jf (X; OM) for some fixed n, and the set of restrictions }f the jueH to Jf(X;(),,) is equicontinuous, by (i) above and (12.15.7.1). Hence the assertion follows from (12.15.7). It remains to show (by the same reasoning as in (12.15.7)) that, when we dentify H with its image L in the product space CNXN by means of the napping /i^«/mw, M»> L is closed in CNXN. If (jik) is a sequence of points >f H such that each of the sequences «/mn, ]"*>)*£ i is convergent, then it bllows from (12.15.7) that, in each Jf(X;On), the restrictions of the fik converge to a continuous linear form. Hence the sequence (/**) converges vaguely to a measure on X. Q.E.D, We shall see later (13.20) that condition (i) in (13.4.2) can also be written n the form |ju|(K) <; CK for each measure ju e H. Notice also that (13.4.2(ii)) implies (13.4.1) as a particular case. In particular: 13.4.3) Let v be a positive measure on X. Then the set of complex measures i such that \IJL\ f$ v is metrizable and compact with respect to the vague topology. For this set is evidently bounded and closed in MC(X) in the vague opology. It should be noted that a bounded set of measures does not necessarily atisfy the hypothesis of (13.4.3), of X, there exists a real number CK > 0